You can try fusion genres of jazz with something you like, and see if it's fun that way. E.g. you've posted about The Lords of the New Church, who seem to have been playing punk and post-punk:

Jazz with post-punk: James Chance & The Contortions

Jazz-mathcore: Needle Play

Instrumental noise-rock/math-rock/punk-jazz: Lanzallamas

I might be able to throw in more names if you name other genres. In particular, I have a bunch of various jazz-metal, e.g. jazz-grindcore/ska: Le Scrawl.